Abstract:There are many papers studying polynomial tractability
for multivariate problems.
Polynomial tractability means that the minimal number
of information evaluations needed to reduce the initial
error by a factor of for a multivariate problem defined on
a space of d-variate functions may be bounded by a polynomial in
and d and this holds for all
.

We propose to study
generalized tractability by verifying when
can be bounded by a power of for
all , where can be a proper subset
of . Here T is a tractability function
which is non-decreasing in both variables and grows slower than
exponentially to infinity.
In this paper we study the set
for some and .
We study linear tensor product problems for which we can compute
arbitrary linear functionals as information evaluations.
We present necessary and sufficient conditions on T such that
generalized tractability holds for linear tensor product problems.
We show a number of examples for which polynomial tractability
does not hold but generalized tractability does hold.